Curve based cryptography found some extra applications in protocols using 
pairings. Even though they are usually stated as using bilinear maps from G_1 
\times G_1 the protocols can also be applied for two different input groups. 
Here one can make use of the definition of the Tate-Lichtenbaum pairing and 
make a clever choice of the residue classes involved in the second argument. 
This leads to a speed-up of the pairing computation. Basically one uses 
divisors with only one point in the support. Such a choice was already 
proposed by Duursma and Lee, however they use it in the first argument and in 
conjunction with distortion maps. We give arguments that these choices are 
actually sound and show how this can be applied on non-supersingular curves 
where one does not have distortion maps.